{"id":227453,"date":"2025-10-20T18:19:10","date_gmt":"2025-10-20T18:19:10","guid":{"rendered":"https:\/\/www.newsbeep.com\/ca\/227453\/"},"modified":"2025-10-20T18:19:10","modified_gmt":"2025-10-20T18:19:10","slug":"you-see-saturns-rings-she-sees-hidden-number-theory-harvard-gazette","status":"publish","type":"post","link":"https:\/\/www.newsbeep.com\/ca\/227453\/","title":{"rendered":"You see Saturn\u2019s rings. She sees hidden number theory. \u2014 Harvard Gazette"},"content":{"rendered":"

Why are there gaps in the asteroid belt?\u00a0<\/p>\n

The answer can be found in dynamical systems, the branch of mathematics that deals with the evolution of chaotic systems over time. The field was the subject of a recent Harvard Radcliffe Institute talk by Laura DeMarco.\u00a0<\/p>\n

\u201cIt turns out there\u2019s a lot of hidden number theory in the planets\u2019 very motion, for example, in the structure of the rings around Saturn,\u201d explained DeMarco<\/a>, Radcliffe Alumnae Professor at Harvard Radcliffe Institute and a professor of mathematics in the Harvard Faculty of Arts and Sciences. There are places asteroids simply cannot be, she went on, because of the underlying number theory that \u201cprohibits certain behavior.\u201d\u00a0<\/p>\n

To find equations that predict real-world systems such as planetary motion, DeMarco researches sequences of numbers that are generated recursively. Perhaps the most best-known example is the Fibonacci sequence, a linear recursion in which each number is the sum of the two numbers that came before it, starting with 0 and 1. Visual representations of the Fibonacci sequences appear in the natural world \u2014 for example, in the scales of pine cones, in pineapple fruitlets, and in the structure of nautilus shells.\u00a0<\/p>\n

\u201cThis suggests that somehow the Fibonacci numbers can be used to encode some aspects of the natural world,\u201d DeMarco said, just like more complex nonlinear sequences show up in the chaotic and seemingly unpredictable structures of planetary motion, weather patterns, and changes in animal populations<\/a>.\u00a0<\/p>\n

\u201cA lot of what I think about may or may not be useful for real life, everyday life,\u201d she said. \u201cBut it\u2019s trying to understand how the complexity of numbers, and the interactions of numbers, force geometry, force certain geometric structures.\u201d<\/p>\n

\"Visualization<\/p>\n

A visualization of the Mandelbrot Set.<\/p>\n

Photo courtesy of Laura DeMarco<\/p>\n

DeMarco is fascinated by the complexity of seemingly simple sequences. For example, every number in the Fibonacci sequence, except 8 and 144, has a prime factor that has not appeared earlier in the sequence.<\/p>\n

\u201cAt every step, you\u2019re going to see at least one new [prime] number, suggesting that there\u2019s something rich and complex about this sequence,\u201d DeMarco said. \u201cAny old sequence is not going to have this feature.\u201d\u00a0<\/p>\n

The theorem that almost every number in certain recursive sequences like the Fibonacci has at least one prime divisor that does not appear earlier in the sequence was first proved in 1913 by the mathematician R.D. Carmichael. What\u2019s not known is whether a similar theorem holds true for nonlinear recursive sequences. Searching for the answer to that question has led DeMarco from the numeric to the psychedelic \u2014 or at least, the visually psychedelic.\u00a0<\/p>\n

\"Visualization<\/p>\n

One of DeMarco\u2019s favorite visualizations looks remarkably like an eye.<\/p>\n

Photo courtesy of Laura DeMarco<\/p>\n

DeMarco graphs nonlinear sequences into complex, recursive visualizations, looking for patterns of symmetry and asymmetry. The asymmetry in her visualizations provides evidence that multiple recursions, when run simultaneously and following very simple sets of rules, can generate extraordinary complexity.<\/p>\n

\u201cI don\u2019t intend to create art, per se, but somehow and sometimes the pictures are surprisingly beautiful,\u201d she said.\u00a0<\/p>\n

One of her favorite visualizations looks remarkably like an eye. It seems perfectly symmetrical, but it contains slight variations, and, when you zoom in, a tiny copy of a famous fractal shape, the Mandelbrot Set.<\/p>\n

Like the Fibonacci sequence, the Mandelbrot Set is defined by a simple rule: It contains all the complex numbers for which a nonlinear sequence, when repeated, does not result in ever-larger values \u2014 that is, it does not diverge to infinity. If the complex number used as a constant yields a result greater than 2, it\u2019s not part of the Mandelbrot Set.\u00a0<\/p>\n

In DeMarco\u2019s eye visualization, the Mandelbrot set shows up on one side but not the other, acting, in her words, as a \u201ccertificate of independence\u201d between the two sides. Those tiny distinctions, invisible to the naked eye, allow her to draw conclusions about the underlying math.\u00a0<\/p>\n

\"Visualization\u201cI don\u2019t intend to create art, per se, but somehow and sometimes the pictures are surprisingly beautiful,\u201d said DeMarco.<\/p>\n

Of course, a good visualization is just the first step. \u201cI need some logical argument to build on what\u2019s known, that\u2019s not just true for what you see in the picture, but that is always true.\u201d<\/p>\n

DeMarco shared more about her work in a 2024 conversation<\/a> on the Harvard Radcliffe Institute\u2019s podcast, BornCurious.<\/p>\n","protected":false},"excerpt":{"rendered":"Why are there gaps in the asteroid belt?\u00a0 The answer can be found in dynamical systems, the branch…\n","protected":false},"author":2,"featured_media":227454,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[49,48,33495,994,66,108681],"class_list":["post-227453","post","type-post","status-publish","format-standard","has-post-thumbnail","category-science","tag-ca","tag-canada","tag-faculty","tag-research","tag-science","tag-space-astronomy"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/227453","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/comments?post=227453"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/227453\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media\/227454"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media?parent=227453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/categories?post=227453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/tags?post=227453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}